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It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every $n\geq4$, does there exist an $n$-manifold with exotic smooth structures?

In other words, for every $n\geq4$, does there exist topological $n$-manifolds which admit more than one diffeomorphism class of smooth structures?

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    $\begingroup$ Are you asking if for every n there exist topological n-manifolds which admit more than one diffeomorphism class of smooth structures? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 18, 2016 at 4:48
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    $\begingroup$ Done! Although 'exotic smooth structure', 'exotic manifold' etc. seem to be standard terms. $\endgroup$
    – timur
    Commented May 18, 2016 at 5:09
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    $\begingroup$ Exotic only makes sense when there is a standard smooth structure, and then asking for exotic structures on random manifolds does not make much sense. In the case of spheres, for example, the term distinctly refers to smooth structures different from in usual one, but there is no "usual one" on a general manifold. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 18, 2016 at 5:13
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    $\begingroup$ A precise question is always better. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 18, 2016 at 5:28
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    $\begingroup$ From a recently arXiv'd paper: "Following results of Moise [35], Kervaire-Milnor [25], Browder [10] and Hill-Hopkins-Ravenel [19], we show that the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$ and $S^{61}$." arxiv.org/abs/1601.02184 This post gives a lucid summary of the results: math.stackexchange.com/a/1609522/155629 $\endgroup$
    – Travis Willse
    Commented May 18, 2016 at 7:27

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Yes. For every $n\ge 5$ there are exotic tori.

In fact, the PL-structures on $T^n$ are in one-to-one correspondence with $H^3(T^n;\mathbb{Z}/2)$, and every one of these is smoothable (Reference: "Surgery on Compact Manifolds" by C. T. C. Wall, Chapter 15A). Since any smooth manifold admits a unique PL-structure up to PL-isomorphism, it follows that there are many manifolds homeomorphic but not diffeomorphic to the standard torus.

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  • $\begingroup$ Thanks a lot! I am sure this builds on contributions from many people, but if I want to attribute this result to a few individuals, who would be most appropriate? $\endgroup$
    – timur
    Commented May 18, 2016 at 16:14
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    $\begingroup$ I'm not an authority on this myself, but search for papers on "homotopy tori" and "fake tori" around 1969-70. I think the main names are Hsiang and Shaneson, Wall and Casson (all of whom apparently proved this result independently using the surgery machine). $\endgroup$
    – Mark Grant
    Commented May 18, 2016 at 16:25
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    $\begingroup$ There are not really as many PL structures on $T^n$ as that: typically one mods out $H^3(T^n;\mathbb{Z}/2)$ by the action of $GL(n,\mathbb{Z}/2)$. This is discussed by Wall in the reference given. Nonetheless, $T^n$ does have multiple PL structures up to PL-homeomorphism, for all $n \geq 5$, so the thrust of the answer is correct. $\endgroup$
    – Dave Davidson
    Commented Mar 27, 2023 at 0:50
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    $\begingroup$ Is the dimension range for the sentence beginning with "In fact" n ≥ 5 ? $\endgroup$
    – Daniel Asimov
    Commented Mar 27, 2023 at 2:31

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